标题： 弗洛凯理论及其在开放量子和经典系统中的应用 显示英文标题

标题： Floquet theory and applications in open quantum and classical systems

摘要： 本文回顾了分析开放（耗散）量子或经典系统中弗洛凯工程（FE）现象的理论方法，重点介绍了我们的最新研究成果。 在许多关于量子系统中FE的理论研究中，研究人员使用了基于薛定谔方程的闭合（孤立）量子系统的弗洛凯理论。 然而，如果我们考虑由振荡场（如激光）驱动的材料中的FE，目标系统与环境（热浴）之间微弱但有限的相互作用是不可避免的。 在本文中，我们通过量子主（GKSL）方程描述这些周期性驱动的耗散系统。 特别是，我们展示了由于驱动场的能量注入与向热浴的释放之间的平衡，在长时间驱动后会出现非平衡稳态。 除了量子系统外，如果我们试图将弗洛凯理论简单地应用于周期性驱动的经典系统，则会失败，因为运动方程（EOM）通常是非线性的，而弗洛凯定理只能应用于线性微分方程。 相反，通过考虑经典变量的分布函数（即福克-普朗克方程），可以得到驱动系统的有效EOM。 我们阐述了经典系统中弗洛凯理论的本质。 在弗洛凯理论基础上，我们回顾了FE的典型例子（弗洛凯拓扑绝缘体、金属和磁体中的逆法拉第效应、卡皮察摆等）以及耗散辅助的FE。 显示英文摘要

摘要： This article reviews theoretical methods for analyzing Floquet engineering (FE) phenomena in open (dissipative) quantum or classical systems, with an emphasis on our recent results. In many theoretical studies for FE in quantum systems, researchers have used the Floquet theory for closed (isolated) quantum systems, that is based on the Schr\"odinger equation. However, if we consider the FE in materials driven by an oscillating field like a laser, a weak but finite interaction between a target system and an environment (bath) is inevitable. In this article, we describe these periodically driven dissipative systems by means of the quantum master (GKSL) equation. In particular, we show that a nonequilibrium steady state appears after a long driving due to the balance between the energy injection by the driving field and the release to the bath. In addition to quantum systems, if we try to simply apply Floquet theory to periodically driven classical systems, it failed because the equation of motion (EOM) is generally nonlinear, and the Floquet theorem can be applied only to linear differential equations. Instead, by considering the distribution function of the classical variables (i.e., Fokker-Planck equation), one can arrive at the effective EOM for the driven systems. We illustrate the essence of the Floquet theory for classical systems. On top of fundamentals of the Floquet theory, we review representative examples of FEs (Floquet topological insulators, inverse Faraday effects in metals and magnets, Kapitza pendulum, etc.) and dissipation-assisted FEs.